Answer
$2-\sqrt 2$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
Thus, the area of the region is:
$A=\int_a^b f(x) \ dx= \int_{0}^{\pi/4} \sin x \ dx+\int_{\pi/4}^{\pi/2} \cos x \ dx \\=[-\cos x]_0^{\pi/4}+[\sin x]_{\pi/4}^{\pi/2} \\=[\dfrac{-1}{\sqrt 2}+1]+[1-\dfrac{1}{\sqrt 2}]\\=2-\sqrt 2$
